Josephson Effects and Unconventional Superconductors

The macroscopic condensate wave function has a definite phase, and a phase is observable only through interference. A thin barrier between two superconductors lets their condensates interfere, and the resulting Josephson effects turn the phase into measurable current, voltage, and flux. The same interference, plus a survey of the cuprate superconductors, closes the module by pointing past the phonon BCS mechanism to pairing that is still not fully understood.

The dc and ac Josephson effects

Two superconductors separated by an insulating barrier a few nanometres thick form a Josephson junction. Label the condensate wave functions ${\psi }_1=\sqrt{n_1}{e}^{i{\theta }_1}$ and ${\psi }_2=\sqrt{n_2}{e}^{i{\theta }_2}$. Across the barrier they are weakly coupled, and the two-state Schrödinger equations read

with coupling energy $K$ and chemical potentials ${\mu }_1,{\mu }_2$. Separating each equation into modulus and phase and writing the phase difference $\delta ={\theta }_2-{\theta }_1$, the rate of pair transfer across the barrier is

the dc Josephson effect: a supercurrent up to $I_c$ flows with no voltage, fixed entirely by the phase difference of the two condensates. The phase evolves according to

since a voltage $V$ across the junction shifts the pair energy by $2eV$. With a constant voltage the phase winds linearly and the current oscillates — the ac Josephson effect,

The frequency depends only on $2e/h$, so a measured frequency fixes a voltage to the precision of a frequency standard; the ac Josephson effect defines the volt.¹

SQUID magnetometry

Put two junctions in parallel on a superconducting loop that encloses a magnetic flux $\Phi$. Flux quantization ties the two phase differences together: going once around the loop, the phase must change by a multiple of $2\pi$, so

The total supercurrent is the sum through the two arms, and with ${\delta }_{1,2}={\delta }_0\mp \pi \Phi /{\Phi }_0$ it becomes

The maximum supercurrent the loop can carry is

an interference pattern periodic in the enclosed flux with period ${\Phi }_0$. A SQUID (superconducting quantum interference device) tracks $I_{\max }$ and so resolves flux changes far smaller than one quantum, giving field sensitivities near $10^{-15}\text{T}$ — the most sensitive magnetometer known, used from geophysics to brain imaging.²

The cuprate superconductors

Until 1986 the highest known $T_c$ was $23\text{K}$. Bednorz and Müller then found superconductivity near $35\text{K}$ in a layered lanthanum copper oxide, and within two years ${\text{YBa}}_2{\text{Cu}}_3{\text{O}}_7$ reached $92\text{K}$, above the boiling point of liquid nitrogen; the mercury cuprates reach $133\text{K}$. These cuprates share a layered perovskite structure in which the current flows in ${\text{CuO}}_2$ planes separated by charge-reservoir layers that dope the planes with carriers.

The undoped parent compound is not a metal but a Mott insulator: strong on-site Coulomb repulsion localizes one hole per copper, and the spins order antiferromagnetically. Adding holes to the ${\text{CuO}}_2$ planes destroys the antiferromagnetism and, past a threshold doping, produces superconductivity. The critical temperature traces a dome as a function of doping, peaking at optimal doping near $0.16$ holes per copper and falling on both the underdoped and overdoped sides.

d-wave pairing and the pseudogap

BCS superconductors have an isotropic s-wave gap: $\Delta (\vec{k})$ is the same sign and magnitude everywhere on the Fermi surface. The cuprates instead pair in a $d_{x^2-y^2}$ channel,

which is largest along the copper–oxygen bond directions, changes sign between them, and has nodes — points where the gap vanishes — along the diagonals. Phase-sensitive experiments, using corner junctions and tricrystal geometries to detect the sign change directly, confirmed the d-wave symmetry. The nodes leave quasiparticles available at arbitrarily low energy, so the heat capacity and penetration depth follow power laws in $T$ rather than the exponential BCS activation.

Above the dome on the underdoped side lies the pseudogap: below a temperature $T^{\ast }$ well above $T_c$, part of the density of states at the Fermi surface is suppressed without full superconductivity. Whether the pseudogap is a precursor to pairing or a distinct competing order remains unsettled, and it is the central open problem of cuprate physics. The pairing glue is widely attributed to antiferromagnetic spin fluctuations rather than phonons, which is consistent with the proximity of the superconducting dome to the antiferromagnetic parent.

Other unconventional families

Superconductivity beyond the phonon s-wave picture now spans several classes.

• Iron-based superconductors (2008), such as the oxypnictide $\text{LaFeAsO}$ and the selenides, reach $T_c$ up to $55\text{K}$ and pair in an $s_{\pm }$ state whose sign reverses between electron and hole Fermi pockets.
• Heavy-fermion and organic superconductors pair near magnetic quantum critical points, again suggesting a magnetic rather than phononic mechanism.
• Hydrogen-rich compounds under pressure, ${\text{H}}_3\text{S}$ and ${\text{LaH}}_{10}$ at megabar pressures, superconduct above $200\text{K}$ by a conventional but extreme phonon mechanism, the light hydrogen mass pushing the Debye energy and the BCS $T_c$ upward.
• ${\text{MgB}}_2$ ($39\text{K}$) is a phonon superconductor with two distinct gaps on different sheets of its Fermi surface.

The Josephson effects show that the condensate phase is a real, measurable quantity, and flux quantization through ${\Phi }_0=h/2e$ recurs in every measurement as the fingerprint of pairing. The cuprates and their relatives keep the pairing state — its symmetry and its glue — an active question, one that the weak-coupling BCS theory frames but does not close.